1 Introduction

We will pick up where we left off in Demo 1 to prepare you for analysis. Although this is not going to be as exhaustive as we would like to do if we were doing a complete project, our goal is to see this project go from messy data $\rightarrow$ prediction & interpretation. After all, we need to give something to the patient advocacy watchdog that they can deploy!

1.1 Our Analysis Plan

Our goal is to see this analysis through to generate a predictive model that the advocacy group can use, and hopefully it will be a decent one. To ensure that it stands the best chance of being a good predictive model, our analysis plan includes all the key steps in the data science life cycle/project flow, including: cleaning, exploration, feature selection, pre-processing, unsupervised machine learning exploration, and supervised modeling. Whew!

Our specific plan is to perform two big analyses in the hopes of having a nice deliverable to present to our client. The first will be a segmentation analysis, or clustering analysis, which will attempt to group our hospitals together on the basis of their attributes and their pneumonia-related readmissions. Thie will be done using a combination of PCA and $k$-means. Our second analysis will be to perform supervised, predictive analysis using an elastic net, which conveniently combines a penalized regression with variable selection so we can point our clients down a path of hospital attributes to focus on, as well as a model they can use to make future predictions.

1.2 Your objective.

Your oject is to work through this Project 1. You will have 31 Questions to answer as you work through this, just as with Project 1. Then, you will be able to choose your own adventure again to perform your own clinical informatics analysis:

Adventure 1.

[Minimal coding, focus is on understanding cleaning, pre-processing, and the ML analyses we performed here.] Choose ONE other condition (anything other than pneumonia) and run through the analyses again, using our same target variable. You will update your question, hypotheses, and predictions, but will be able to re-use all of the other code with very minor modifications. Credit will be based more on interpretation and not on coding. See more information on Adventure 1 here.

Adventure 2.

[More coding, but not as much, with a focus on the steps we’ve undertaken here and extending it to a new question & condition.] Choose a slightly more complicated analysis to undertake, for example, focusing on surgical interventions (HIP-KNEE & CABG) or heart-related conditions (HF, AMI, & CABG). Coding should still be fairly minimal, but you are likely to run into problems with the code working exactly as-is, especially during cleaning. The advantage of this analysis is that it will be much more robust, have larger sample sizes, and would allow us to be able to say something far more informative to a subset of patients (e.g., those considering undergoing surgical interventions). See more information on Adventure 2 here.

Adventure 3.

[Most coding, but no cleaning or pre-processing, with an emphasis entirely on the machine learning analyses.] Continue with the pneumonia-related data that you have here, completely cleaned and pre-processed, and do two more unsupervised OR supervised analyses. For example, you could do another kNN using the $k$ we identify in our Segmentation Analysis here, or you could do a random forest to account for any possible non-linearity. You could even choose to do a Ridge and LASSO regression and compare those analyses to what we get out of an Elastic Net. Any of these are fine, but you must explain your decision in your submission. See more information on Adventure 3 here.

1.3 A Fork in the Road!

This is the first of your choices in your adventure, and what you choose here could either lead to success or… sudden death (dum, dum, dummmm…). Choose wisely, young data scientists!

Option 1.3.1

Load data from the end of Demo 1, which includes the ordinal and frequency encoding you did (plus a little bonus cleaning to help).

Load my version of the fully cleaned, encoded, & ready-to-proceed dataset from the end of Demo 1. These data are not yet ready for analysis (!) but we would be picking up with that here.

## Uncomment if this is your choice:
# load("FY2024_data_files/pneumoniaAnalyzeFullyEncoded2024.Rdata")
# dat2Analyze <- pneumoniaAnalyzeFullyEncoded

Option 1.3.2

Load data from Demo 1 without ANY encoding, but all other tidying the same.

## Comment out if this is NOT your choice:
load("FY2024_data_files/pneumoniaAnalyzeNoEncoding2024.Rdata")
dat2Analyze <- pneumoniaAnalyzeNoEncoding

Question 1: [1 point]

Which dataset do you choose and why?

1.4 Helper code: Using `R` like an object-oriented language

Many of you still may be early enough in your programming journeys that you haven’t yet learned how to fully leverage the objected-oriented nature of programming languages like Python. R is not technically an object-oriented language in the same way, but we can hack it to function like one. That’s exactly what we are going to do here by loading a helper function directly from the source code.

We went through all the trouble in the Demo to perform encoding and now we get to use it! You will use the source() function to load a helper function that I have written, using my answer to the ordinal encoding question of the Demo, so that you can do all your ordinal encoding in a single line of code. Neat, huh?

BUT WHY are we doing the encoding now if we just chose to bring in non-encoded data? Because we can do ordinal encoding before or after data partitioning with impunity. It does not introduce data leakage to do so here, so we can do it in either order. This means that we can do it now simply for convenience.

NOTE: If you chose Option 1, the fully-encoded dataset, you can still run this chunk because there is a conditional in here that will not run your dataset since it is already encoded! But still make sure you understand the idea of source code because we will keep using it throughout the course!

## This calls the code in the associated .R file
source(file = "doOrdinalEncoding.R")

## This checks to see if you loaded the already encoded dataset
if(!exists("pneumoniaAnalyzeFullyEncoded")) {
    ## This sets a list of PATTERNS to match for the columns to ordinal encode
    encodeList <- c("ComparedToNational_", 
                    "PaymentCategory", 
                    "ValueOfCareCategory",
                    "Score_Emergency department volume")

    ## This runs the function, which takes 3 arguments
    ## Open the source code to see what each argument does!
    dat2Analyze <- doOrdinalEncoding(df = dat2Analyze,
                                     encodeList = encodeList,
                                     quiet = FALSE)
}

2 Exploratory Data Analysis: Which of the possible target (outcome) variables should we use? (Continued)

NOTE: What follows is my (long-winded) answer to Question 19 from the Demo.

2.1 Overview of our options

I am going to quickly break down the differences between the possible targets so we can make sure we really understand what they are measuring to allow us to make our most-informed selection of a target. Note: More information was found in this PDF. This is really important for helping us choose the right one!

Notice that I am also including a variable we didn’t look at previously called ComparedToNational_Hospital return days for pneumonia patients. This is because this is our best assessment of how the hospital is performing with regard to pneumonia readmissions relative to a national average. However, even though I’m including it in the table, I think it’s better to think of it as a solid predictor. It’s telling us how length of stay (LOS) in the hospital after pneumonia readmission stacks up compared to the national average. After a hospital readmission, it’s generally better to have fewer return days rather than more because it suggests better quality of care, more timely follow-up, and a greater chance of the patient recovering well at home. Since these are readmissions, having more days than average could also suggest that the hospital missed the severity or didn’t properly plan for the patient’s recovery at home and released them too early.

You may be wondering - could we use it as a target? Sure - but it’s not for the question we were posed by our stakeholder! Note that, if we did, we would need to perform a multinomial classification analysis. So, instead, let’s use this category to help us contextualize our hospitals as we choose between Predicted, Observed, and Expected readmission rates, as well as Excess Readmission Ratio.

Table 1. List of possible target variables from the pneumonia-related hospital readmission data. Which one to choose?
Possible Target	Description
ComparedToNational_Hospital return days for pneumonia patients	Hospital return days measures the number of days patients spent back in the hospital (in the emergency department, under observation, or in an inpatient unit) within 30 days after they were first treated and released for pneumonia. Reported as compared to the national average, such that ‘below average’ is better and ‘above average’ is worse.
ExpectedReadmissionRate	The expected number of readmissions in each hospital is estimated using its patient mix and an average hospital-specific intercept. It is thus indirectly standardized to other hospitals with similar case and patient mixes.
PredictedReadmissionRate	The number of readmissions within 30 days predicted based on the hospital’s performance with its observed case mix. The predicted number of readmissions is estimated using a hospital-specific intercept, and is intended to reflect the annual expected performance of the hospital given its historical case and patient mix and performance.
ExcessReadmissionRatio	The ratio of the predicted readmission rate to the expected readmission rate, based on an average hospital with similar patients. Performance is compared against a ratio of one, such that below one is better and above one is worse in terms of readmission rates.
observed_readmission_rate	Our calculation of the observed number of pneumonia-related readmissions within 30 days found by dividing the number of readmissions observed for the hospital during the given period by the number of discharges for that same period, and multiplied by 100 to put it on the same scale as the Predicted and Expected Readmission Rates. It reflects a true observation as reported by the hospital during that period, but is not adjusted for case mixes or prior information for that hospital. Thus, this is a crude, unadjusted value.

2.2 Summary statistics of the possible targets

Now, I know I didn’t ask you to do it, but I will actually begin my assessment of which target is most appropriate by constructing a quick summary table. In my assessment of what is an appropriate target,

**Table 2. Possible target measure summaries**
Characteristic	N = 4,816¹
National Comparison Return Hospital Days
Better than average	402 / 4,816 (8.3%)
Same as average	2,285 / 4,816 (47%)
Worse than average	827 / 4,816 (17%)
Unknown	1,302 / 4,816 (27%)
Predicted Readmission Rate	16.46 (1.93), 16.27
(Missing)	2,090
Observed Readmission Rate	17.1 (3.7), 16.9
(Missing)	2,603
ExcessReadmissionRatio	1.00 (0.06), 1.00
(Missing)	2,090
Expected Readmission Rate	16.43 (1.51), 16.39
(Missing)	2,090
¹ n / N (%); Mean (SD), Median

We can see right away that we lose nearly 600 more data points if we opt to use our calculated observed_readmission_ratio (NOTE: nearly 500 if using the 2025 FY data), which is going to be a by-product of the fact that the Predicted and Expected readmission ratios are using prior information and information about patient mixes to generate these values. Our observed_readmission_ratio is crude and unadjusted any way - which could present problems when it comes to true comparability. So, off the bat it feels as if the constructed target is possibly not a good choice despite its simplicity for stakeholders to understand.

2.3 Comparing the four possible metrics to`ComparedToNational_Hospital return days for pneumonia patients`

Although ComparedToNational_Hospital return days for pneumonia patients is very compelling, it’s better as a predictor rather than a target because it gets at an aspect of hospital performance: once readmitted due to pneumonia, how long (as compared to the national average) are patients spending in the hospital? This means it makes a really nice predictor to use to compare our four possible choices for pneumonia-related hospital readmissions. We are looking for a clear separation in our ideal target between hospitals that have lower readmission rates AND also tend to do better than the national average at healing their readmitted patients!

So, this makes a really nice variable I could choose to facet plots, which means creating sub-plots that show up as panels. The thing is, if I know I want to facet plots, the first thing I have to do is reshape the data I want from wide into long format using, for example, the pivot_longer() function (the complement of pivot_wider() that we used in the Demo).

I don’t have to store it as a separate data frame (i.e., I can just pipe it directly into ggplot()!) but here I am choosing to so we can more easily make sure it’s reshaped correctly along the way. Thus, I am making a long-format version of the possible target variables called longTargets so that I can better decide which target to choose.

I mentioned violin and density plots specifically in Question 19 of Demo 1, as well as faceting. I am going to show you two ways to explore these data with these types of plots and you can decide which one(s) you find most informative here. You also may have thought of a better way to display the data too!

2.3.1 Example with `geom_violin()`:

Here I am leaving the NA data (which I relabelled as Unknown) to get a sense for the degree of missingness.

Figure 1. Readmission Rates vs. readmitted hospital LOS

Question 2A: [0.5 points]

If you have never encountered violin plots before, they are honestly a wonderful way to assess the symmetry and skew of a distribution while more easily enabling comparisons between groups. They’re effectively a hybrid between a histogram and a boxplot, and we can even opt to overlay the boxplot quartiles in the plots as well, which is particularly handy!

Your task is to do just that by adding geom_boxplot(width = 0.1) (yes, just that one line!) in the appropriate place. Also change include = FALSE to include = TRUE in the R chunk header so that it shows up when you knit!

Question 2B: [0.5 points]

Now that you can compare these groups more easily, what we’re looking for is a balance between clear difference between hospital ordinality relative to the national average (i.e., “Are better hospitals really better in pneumonia readmission rates?”).

Which target(s) appear the most distinct to you? > Your answer here.

2.3.2 Example with `geom_density()`:

We could also choose to overlay density plots instead of violin or boxplots, as this can sometimes help us see subtler separation of groups than we can see otherwise. This time, I am also going to ignore the “Unknown” (i.e., missing) class of hospitals to prevent things from getting too messy!

Question 3: [0.5 points]

Does this view confirm or change your decision from Question 3?

2.4 What is the relationship of pneumonia-related hospital readmissions to the hospital performance metrics?

Something else we could do to help us make a decision is compare across predictors to look for the most striking relationships among the proposed target (readmission rate), how well compared to national average hospitals are performing AFTER readmitting pneumonia patients (how long after readmission they are in the hospital), and relationships among other hospital-performance predictors, like Medicaid spending or pneumonia-related deaths.

So, to do this, I’m going to make another long-format dataset and use this to make a bunch of faceted scatterplots where I put one of the readmission rates (let’s started with Predicted Readmissions) and look for strength in relationships. Note that doing it this way, through plotting, is a heuristic way of choosing a target variable and it isn’t the only way to do it!

Question 4: [1 point]

Change the y-axis out for _each of the four readmission rates. Predicted Readmission Rates has been started for you. Re-run the chunk and compare the results. FOR FULL CREDIT you must leave the graph on the choice you feel most strongly about AND update the title of the graph with which of the four possible readmission rates you chose. Explain below why you chose this one.

HINT 1: What does higher medicare spending per patient tell you about the underlying population of a hospital?

HINT 2: Venous thrombolisms and deep vein thromboses (DVTs) are often “sneakier” conditions that have “silent” symptoms until the patient is in crisis. What would having higher scores at providing prophylaxis for these conditions potentially suggest about a hospital’s performance?

Figure 4. Readmission rate by hospital performance and readmitted patient LOS

HINT 3: Here’s how I would interpret the relationship between PREDICTED pneumonia readmissions and pneumonia death rates, looking at whether hospitals performed better, same, or worse in terms of length-of-stay after readmission. What I notice immediately is a moderate slope for hospitals with worse readmitted lengths of stay (LOS) [maroon line] compared to same [blue line] and especially better [yellow line]. This relationship is negative, implying that as the pneumonia-related death rate increases, readmissions goes down. Oh! So what this implies is that hospitals with higher deaths from pneumonia tend to have lower readmissions - and this is especially true for the ones who are have longer (worse than average) readmitted patient stays. In other words - these hospitals are __ probably releasing their pneumonia patients too soon from their first hospitalization__, leading to higher readmissions, longer readmitted stays, and even higher pneumonia-related deaths!

2.5 What is the relationship of pneumonia-related hospital readmissions to the patient satisfaction metrics?

Now let’s do a similar analysis but for the HcahpsLinear... variables. The HCAHPS (Hospital Consumer Assessment of Healthcare Providers and Systems) is a standardized survey that measures adult inpatient perceptions of hospital care. It’s a national, publicly reported survey mandated by the Centers for Medicare & Medicaid Services (CMS). Because it focuses on aspects of hospital care that are important to patients, it is often used to improve patient satisfaction and hospital quality.

Question 5: [1 point]

Again, you task is to change the y-axis out for _each of the four readmission rates. Predicted Readmission Rates has been started for you. Re-run the chunk and compare the results. FOR FULL CREDIT you must leave the graph on the choice you feel most strongly about AND update the title of the graph with which of the four possible readmission rates you chose. Explain below why you chose this one.

HINT 1: What does higher nurse and doctor communication mean for improved patient outcomes, including readmission?

HINT 2: Why might cleanliness not have much effect here even though pneumonia is a communicable disease?

Figure 5. Readmission rate by patient satisfaction and readmitted patient LOS

HINT 3: Here’s how I would interpret the relationship between PREDICTED pneumonia readmissions and overall hospital rating, looking at whether hospitals performed better, same, or worse in terms of length-of-stay after readmission. What I notice immediately is a pretty substantial negative slope for hospitals with worse readmitted lengths of stay (LOS) [maroon line], moderate negative slope for hospitals with the same readmitted LOS [blue line], and slightly decreasing slope for hospitals with better readmitted LOS better [yellow line]. So what this implies is that hospitals with high ratings tend to have lower pneumonia-related readmissions, even among the hospitals with worse readmitted length-of-stays. This relationships is strongest when hospitals have longer LOS; maybe having a long readmission LOS stay actually implies a form of quality, like “Well, we got it wrong the first time, let’s make darn sure we get it right the second?” But hospitals that were already getting it pretty right don’t have as steep of slope because they were already doing well at readmissions to begin with, hence the almost flat line.

Question 6: [2 points]

What relationships do you find most compelling from these scatterplots and why? Also, name one outstanding question you have at this point, whether it is about the data, the data collection, or something more speculative.

Although there is certainly more EDA we could do, we are going to wrap that up here for the sake of moving on - but, as always, I encourage you to think about or even explore what else you would like to know about your data that we did not do!

2.5 Final remarks on target selection.

Going forward, I have selected PredictedReadmissionRate as our target. In a more thorough analysis, we would likely want to consider:

Circling back to our stakeholder to try to refine our question or target using domain knowledge
Perform experiments with our models to find the best predictive model since that is our ultimate goal.

However, given that we do not have the option or time for that, let me explain my choice. It’s the best balance of distinguishing hospitals with better and worse readmitted hospital stays, while still providing an adjusted, hospital-specific measure. Adjusted values will generally always be a better choice than the crude, even if our crude measure captures something interesting about the data, as it did here, or if it’s easier for our stakholder to comprehend. Our crude measure, observed_readmission_rate did the best job of recapitulating the readmitted hospital LOS (ComparedToNational_Hospital return days for pneumonia patients), but if that’s really our focus, wouldn’t it then be better to switch from a readmission rate to that as our target? Probably.

If you still feel unsure about what is the best choice, you’re not alone - so do I! This is where, because we are working with an imagined stakeholder, we must make the best decision we can given the question we defined in the Demo. I would defend my choice of Predicted Readmission Rate for three key reasons:

It has been adjusted to be hospital-specific, using a hospital-specific intercept and historic data about the hospitals’ performances
We don’t lose nearly 600 data points to missingness.
Our imagined stakeholder, and the question we set, was about readmission rate so even though readmitted return hospital days might also be a similarly viable candidate, it doesn’t reflect our imagined stakeholder’s request. Note that the ExpectedReadmissionRate is standardized for similar hospitals, and may not truly reflect a given hospital; the ExcessReadmissionRatio, while informative, was the least likely to reflect the distinctiveness in hospital return days, which is still of interest to us.

But remember that what matters most is that we are matching our question.

Question 7: [1 point]

Do you agree with my choice of target? Why or why not? (You are free to disagree!)

2.6 Bias and Limitation

Question 8: [2 points]

All studies have bias (introduced or systemic error) and limitations (failure to fully explain something). We could go more deeply into these concepts, but for now I want you to take a stab and just brainstorming either ways we might have bias OR the limitations of using this target variable. You do not need to answer both, but you’re welcome to do so. You can read more about bias here or see some deeper explanations of limitations here. You do not need to write a lot; just 2-3 sentences. I am simply asking you to pause and reflect on our choices here before we proceed.

3 Pre-processing and Feature Selection

The time has finally come to officially move back into pre-processing and feature selection so that we can finally build some models. I want you to take note of the order in which I’m doing things here; the flow can change from project to project, but generally it’s a good idea to get your dataset to a state of manually curated features before making your data partition.

3.1 Further feature selection and refinement.

3.1.1 Dropping the targets we aren’t planning to use. [Manual Method]

It’s time to say goodbye to the features we know would be far too redundant or non-informative given our target of PredictedReadmissionRate. We don’t need any of the Sample_ columns, for example; these are merely sample sizes. Useful if we were doing standardization, but not necessary for us right now. We also are dropping the other possible target features that are too redundant with PredictedReadmissionRate.

Question 9A: [0.5 points]

Add your comments to the code as indicated to explain what I was doing at each of the steps.

dat2Analyze <- dat2Analyze %>% 
  ## COMMENT HERE
  select(-ExpectedReadmissionRate, 
         -ExcessReadmissionRatio,
         -observed_readmission_rate, 
         -contains(c("Sample_", "NumberOfPatients")),
         -NumberSurveysCompleted) %>% 
  ## COMMENT HERE
  rename(`Median time (minutes) patients spent in ED` = 
         `Score_Average (median) time patients spent in the emergency department before leaving from the visit A lower number of minutes is better`,
         `Composite patient safety` = `Score_CMS Medicare PSI 90: Patient safety and adverse events composite`,
         `ComparedToNational_Composite patient safety` = `ComparedToNational_CMS Medicare PSI 90: Patient safety and adverse events composite`) %>% 
  ## COMMENT HERE
  select(PredictedReadmissionRate, everything())

## COMMENT HERE
colnames(dat2Analyze) <- gsub('Score_|HcahpsLinearMeanValue_|_Payment', '', colnames(dat2Analyze))

3.1.2 Dropping any features with near-zero variance. [Automated Method]

Next, we are going to take advantage of the nearZeroVar() function that is part of the caret package. We will drop any of the columns with near-zero variance. This is a way of automating (and thus standardizing) the process for choosing to drop non-informative columns due to zero or nearly zero variance.

Question 9B: [0.5 points]

If you are not familiar yet with the concept of zero or near-zero variance, do a little digging. Based on what you read, try to explain for your naive advocacy group stakeholders, why it’s important to remove the features with little variance.

Table 3. Columns dropped due to near-zero variance.
Variables Dropped
ComparedToNational_Postoperative respiratory failure rate
ComparedToNational_Perioperative pulmonary embolism or deep vein thrombosis rate

Take note of the number of features here.

Question 9C: [0.5 points]

You may or may not have noticed that I moved FacilityID to our rownames in the datasets you loaded today. Why did I choose to do that? Why do I want to make sure unique identifiers, like FacilityID, are out of the dataframe before I either (a) run nearZeroVar() or (b) proceed with the rest of the analysis?

Question 9D: [1 point]

Do some research on, or lean into your previous knowledge, about the curse of dimensionality or the big-P, little-N (n >> P) problem. Using whatever you find or remember, can you justify why we likely do NOT have a big-P, little-N (n >> P) problem here? Make sure to explain.

3.2 Assess missingness & devise an imputation plan

We are going to deal with imputation in a rather quick and elegant way that requires minimal effort from us using a model-based approach with missForest. You may recall from lecture that I said we are choosing a model-based approached without uncertainty because our ultimate goal is to create a deployable model for our stakeholder.

Question 10: [1 point]

Write some code that will quickly tally the number of columns in the dataset that contain missing values. You will find colSums() and is.na() functions likely useful here. What percent of the features have at least some level of missingness?

# Your code here.

Which column(s) are not missing any data?

How many complete observations do we have in the dataset? You may find nrow() with complete.cases() or drop_na() useful!

# Your code here..

3.2.1 Use the `md.pairs()` function from the `mice` package to assess the extent of missing values before imputing.

When we are missing a lot of data, as we are here, it can sometimes be useful to predetermine whether the patterns of missing correlate with our target. Why? We care if missing values correlate with the target because this may indicate that the missingness itself carries information. Thus, ignoring it could lead to biased models or low predictive power!

In fact, missingness can itself become a predictor. If a missing value is associated with the outcome, it may carry predictive signal. For example, in our data, hospitals with high rates of missingness in Left before being seen highly associated with pneumonia readmission might indicate a worrying pattern. Perhaps hospitals are negligently failing to report how many of their patients leave without being seen because of larger, systemic issues that ALSO happen to impact overall quality and patient care. Thus, missingness could itself become a predictor!

To do, I will leverage the md.pairs() function from the mice package which calculates the pairwise correlations of missingness between variables and allows us to visualize those patterns using a clustered heatmap. When I do this, I will drop State from the matrix because it will create errors if I retain it. (Why?)

## Drop State and analyze the pairwise missingness using the mice package
miceMatrix <- dat2Analyze %>% select(-State) %>% md.pairs()

What md.mice() does is create two pairwise correlation matrices, mr and mm, where:

mr = Missing vs. Responded
- Pairwise matrix showing how often the row has a missing value in one variable and an observed (i.e., non-missing or “responded”) value in the other
mm = Missing vs. Missing
- Pairwise matrix showing how often the row has a missing value in one variable and missing value in the other variable

Let’s take a look at the first 6 rows and 4 columns of the mr matrix:

Table 4. Missing vs. Reponded Matrix Slice
	PredictedReadmissionRate	MRSA Bacteremia	Payment for pneumonia patients	Abdomen CT Use of Contrast Material
PredictedReadmissionRate	0	48	630	1238
MRSA Bacteremia	1041	0	1650	2236
Payment for pneumonia patients	21	48	0	661
Abdomen CT Use of Contrast Material	37	42	69	0
Death rate for pneumonia patients	10	47	6	609
Postoperative respiratory failure rate	425	59	1042	1574

Any given cell in a matrix is always of the form $[i, j]$, where $i$ is the row and $j$ is the column - just as you are used to with dataframes! So, in these correlation matrices of our pairwise variables, cell $[1,1]$ is PredictedReadmissionRate vs. itself, so the count is 0 - as is every value on the diagonal, like with all correlation matrices. For cell $[1,2]$ PredictedReadmissionRate vs. , there are 42 observations where variable PredictedReadmissionRate ($i$) is missing but MRSA Bacteremia ($j$) was observed. Thus, we can extract the number of cases where variable $i$ is missing and variable $j$ is observed using miceMatrixmr[i, j].

Now, let’s take a look at the first 6 rows and 4 columns of the mm matrix now:

Table 5. Missing vs. Missing Matrix Slice
	PredictedReadmissionRate	MRSA Bacteremia	Payment for pneumonia patients	Abdomen CT Use of Contrast Material
PredictedReadmissionRate	2090	2042	1460	852
MRSA Bacteremia	2042	3083	1433	847
Payment for pneumonia patients	1460	1433	1481	820
Abdomen CT Use of Contrast Material	852	847	820	889
Death rate for pneumonia patients	1292	1255	1296	693
Postoperative respiratory failure rate	1788	2154	1171	639

Similarly, miceMatrix$mm[i, j] gives us the number of cases where both variables $i$ and $j$ were co-missing.

Question 11A: [0.5 points]

Explain what you think the number in cell $[6,1]$ means. Out of all the observations in the dataset, does this feel like a high degree of co-missingness?

Lastly, we can calculate a proportion of missingness vs. response between any two variables $i$ and $j$ such that:

$Proportion_{i,j} = \frac{mr_{i,j}}{mr_{i,j} + mm_{i,j}}$

In non-mathy speak, it’s the fraction of the times that variable $j$ is observed among rows where $i$ is missing. So, if you want to impute variable $i$ - let’s say it’s MRSA Bacteremia - then you will be interested in the variables typically observed (present in the dataset) when $i$ is missing. Those are the variables that will be able to help us predict, or fill in, $i$.

If this ratio is close to 1, $j$ is nearly always available when $i$ is missing, so $j$ might be useful for imputing $i$.
If this ratio is close to 0, $j$ is usually missing when $i$ is missing, so it’s not helpful for imputing $i$. You can also use this to tell you about patterns of missingness with regard to your target variable when you target is $i$.

Figure 5. Correlation between Missing and Response

It may feel a little counterintuitive, but here darker colors indicate a higher correlation between missingness and response (proportions nearer to 1). In other words, these are the variables that are going to be more informative during imputation for a given variable.

But what if we wanted to know about patterns of missingness with regard to our target, PredictedReadmissionRate? Well, we could get that too!

Figure 6. Correlation between Missing and Missing for Target

Question 11B: [0.5 points]

What columns seem to be most correlated in terms of missingness to our target, PredictedReadmissionRate? Does this give you any cause for concern? Feel free to speculate as to why you might see this.

3.3 Drop rows that are missing from the target variable.

We will not be able to use those rows at all moving forward, sadly. That is going to reduce our sample size pretty sizably, from $N = 4,816$ to only $N = 2,726$ (or from $N = 4,775$ to only $N = 2,731$ .

## [1] "Rows: 2726"  "Columns: 31"

Question 12: [0.5 points]

Thinking back to your answer to Question 12, do you feel any differently given these dimensions of the dataset?

3.4 Do the splits!

At this stage, we are almost ready to impute - but not until we have done our data partitioning first! In fact, splitting our data into training and testing partitions first is really important to prevent data leakage.

Question 13: [1 point]

Explain to your stakeholder, in simpler language, (1) why we partition data into testing and training sets and (2) what data leakage is. You do not need to go into great detail, but it should explain enough that they understand the basic ideas.

You may recall that I mentioned that the optimal ratio is has been argued to be $\sqrt{p} : 1$, where $p$ is the number of parameters (which may or may not equal your predictors depending on your planned analysis). Note that you do not have to split only a single time; you could make different splits for different analyses, if needed. However, we are going to use a single split here for convenience.

Now, wouldn’t it be nice to have a handy function that could calculate the optimal split ratio for us? So, let’s write one! This is something you can use with ANY analysis going forward!

Question 14A: [1.5 points]

There are three parts to this question to get it to all come together correctly.

Comment the function to explain what it does (I’ve defined the arguments for you)
Run the function (fill in the blank)
Fill in the missing piece in the data partitioning chunk below

calcSplitRatio <- function(p = NA, df) {
  ## @p  = the number of parameters. by default, if none are provided, the number of columns (predictors) in the dataset are used
  ## @df = the dataframe that will be used for the analysis
  
  ## COMMENT HERE
  if(is.na(p)) {
    p <- ncol(df) -1   ## COMMENT HERE
  }
  
  ## COMMENT HERE
  test_N <- (1/sqrt(p))*nrow(dat2Analyze)
  ## COMMENT HERE
  test_prop <- round((1/sqrt(p))*nrow(dat2Analyze)/nrow(dat2Analyze), 2)
  ## COMMENT HERE
  train_prop <- 1-test_prop
  
  ## COMMENT HERE
  print(paste0("The ideal split ratio is ", train_prop, ":", test_prop, " training:testing"))
  
  ## COMMENT HERE
  return(train_prop)
}

## Fill in the blanks to run:
# train_prop <- ___(df = ___)

Hint: One important note here is that if the number of parameters are not provided (which is true for the starter code I gave you to run it - it does NOT pass any parameters to that argument!), then by default our function is designed to take the number of columns of the dataframe and subtract one for the target variable. Neat, huh?

Now, uncomment this code and fill it out to run it.

# ind <- createDataPartition(dat2Analyze$___,   ## Put the target here!
#                            p = ___,           ## Use the object you just made!
#                            list = FALSE)
# 
# train <- dat2Analyze[___, ]
# test <- dat2Analyze[-c(ind), ]

Question 14B: [0.5 points]

What split ratio did you get? Is it close to the canonical 80-20 split? Why do you think that happened?

Did you get stuck somewhere? If so, load the test and train data in case you struggled with Question 18 so you can proceed:

load(file = "FY2024_data_files/pneumoniaTrain.Rdata")
load(file = "FY2024_data_files/pneumoniaTest.Rdata")

3.5 Impute missing variables using `missForest`.

We are going to take advantage of the missForest() function from the missForest package. In a nutshell, what missForest does is it will fit either a regression- or classification-based random forest model and use the OOB (“out-of-bag”) results to predict and fill in NAs. The advantage is that this process is non-linear, done in just a few lines of code, and can handle mixed data-types (although you need to make sure that all your numerical types are of the same numerical type, and that all characters or factors are of the same type).

3.5.1 Running `missForest`

The way I have chosen to do this is to separate the data so that I can turn the encoded categories back into categories temporarily, then stitch them back together into a temporary dataframe so that I can run missForest. Notice that I temporarily drop the State and target features; I drop State to spare ourselves a little computational time and I drop PredictedReadmissionRate to prevent data leakage. Be patient!!. Since this is randomForest, it could take a minute or two to run. NOTE: If your machine fails to run this, there are data you can load. Some computers may struggle to run this.

# Remove encoded categorical variables for imputation, plus the target and state (which have no missing)
data_sans_encoded = train %>% 
  select(-PredictedReadmissionRate,
         -State,
         -`Emergency department volume`, 
         -contains("ComparedToNational"))

data_cat = train %>% 
  select(`Emergency department volume`, 
         contains("ComparedToNational")) %>% 
  mutate_all(as.factor)

# Stitch them back together:
temp <- cbind(data_sans_encoded, data_cat)

# Impute missing values using missForest
imputedTrain <- missForest(temp, 
                          variablewise = TRUE,
                          verbose = FALSE)

NOTE: Did you catch that I not only removed the target before imputation, but I dropped State too? This is because we know State is/will be frequency encoded (depending on our path), and it could create imputation issues as a result. This leaves me with 29 columns to perform imputation on.

3.5.2 Load this if it will not run for you

load(file = "FY2024_data_files/imputedTrain.Rdata")

3.5.3 Quick summary of the results

NOTE: If yours didn’t run, make sure to look at the table output in the knitted HTML I provided you. Also, change the include=FALSE in the chunk header if you were not able to get missForest to run.

Table 6. missForest OOB Error Rates for the imputed variables, training dataset
OOB Error	Variable
MSE
0.28	MRSA Bacteremia
948452.73	Payment for pneumonia patients
17.67	Abdomen CT Use of Contrast Material
3.72	Death rate for pneumonia patients
6.81	Postoperative respiratory failure rate
0.45	Perioperative pulmonary embolism or deep vein thrombosis rate
0.01	Composite patient safety
0.00	Medicare spending per patient
291.21	Healthcare workers given influenza vaccination
5.08	Left before being seen
124.19	Venous Thromboembolism Prophylaxis
0.95	Nurse communication
1.81	Doctor communication
4.52	Staff responsiveness
4.36	Communication about medicines
4.69	Discharge information
1.25	Care transition
11.94	Cleanliness
14.26	Quietness
0.90	Overall hospital rating
2.37	Recommend hospital
0.00	SurveyResponseRate
PCF
0.09	PaymentCategory for pneumonia patients
0.46	Emergency department volume
0.13	ComparedToNational_MRSA Bacteremia
0.06	ComparedToNational_Death rate for pneumonia patients
0.04	ComparedToNational_Composite patient safety
0.36	ComparedToNational_Hospital return days for pneumonia patients

Question 15: [1 point]

What does MSE (Mean Squared Error) and PCF (Proportion of Falsely Classified) indicate here? You may find looking at the missForest Vignette helpful. Which variable(s) had the highest OOB error?

NOTE 1: If the OOB error (either MSE or PCF) from missForest is really high, it’s a red flag that the imputation for those variables is untrustworthy. This could be do to too much missingness, weak relationships among the variables making it hard to predict, or low quality data (e.g., noisy, sparse). In such cases, we want to DROP any variables with extremely high OOB error because they add noise. A general rule of thumb is that if the PCF > ~0.3 (30% of the imputed values are wrong) or if there is especially high MSE (inflation) relative to the variance of the variable, we should drop it. Let’s set those aside so we can drop them from test and train both downstream!

NOTE 2: Even though the PCF is a bit too high, I am choosing to retain ComparedToNational_Hospital return days for pneumonia patients because it is a primary predictor. This could be a poor choice, however! Every decision we make as data scientists can come with pitfalls.

## These look okay. Uncomment to see their variances.
#var(train$`Hospital return days for pneumonia patients`, na.rm = T)
#var(train$`Median time (minutes) patients spent in ED`, na.rm = T)
#var(train$`Healthcare workers given influenza vaccination`, na.rm = T)

cols2drop <- c("Payment for pneumonia patients", "Emergency department volume")

Question 16: [1 point]

Alternatively, we could use MICE (Multivariate Imputation with Chained Equations), which is a method that will model uncertainty along with the variables. Do a little research. How does MICE work? Include a citation to a source.

3.5.4 Putting it alllllll back together…

We are almost done with imputation. The last thing we need to do is extract the imputed values from imputed_data$ximp, and add the State and target variable back. Then we get to wash, rinse, and repeat for the test set! Oh my!

3.5.4.1 Put the original `State` and `target` variables back

imputedTrain <- imputedTrain$ximp %>% 
  ## Put the original State and target variables back on
  mutate(State = train$State,
         PredictedReadmissionRate = train$PredictedReadmissionRate)

3.5.4.2 Now do the imputation for `test`

# Remove encoded categorical variables for imputation, plus the target and state (which have no missing)
data_sans_encoded = test %>% 
  select(-PredictedReadmissionRate,
         -State,
         -contains("ComparedToNational"))

data_cat = test %>% 
  select(contains("ComparedToNational")) %>% 
  mutate_all(as.factor)

# Stitch them back together:
temp <- cbind(data_sans_encoded, data_cat)

# Impute missing values using missForest
imputedTest <- missForest(temp, 
                          variablewise = TRUE,
                          verbose = FALSE)

3.5.4.3 Load this if it will not run for you

load(file = "FY2024_data_files/imputedTest.Rdata")

3.5.4.4 Put the original `State` and `target` variables back on `test`

imputedTest <- imputedTest$ximp %>% 
  ## Put the original State and target variables back on
  mutate(State = test$State,
         PredictedReadmissionRate = test$PredictedReadmissionRate)

3.5.4.5 Drop the features we determined we should not retain due to unstable imputation from BOTH `train` and `test`

imputedTrain <- imputedTrain[, !(names(imputedTrain) %in% cols2drop), drop = FALSE]
imputedTest  <- imputedTest[, !(names(imputedTest) %in% cols2drop), drop = FALSE]

Question 17: [1 point]

Quickly explore how well the imputation did by choosing at least one numeric variable and one of the categorical variables. Did the distributions or frequencies change drastically? You may find the summary() and table() functions to be fastest here.

# Your code here.

3.6 Another Fork in the Road

It’s time, yet again, to make a choice. If you opted to load the fully encoded dataset and everything has been done on that, you will run the first chunk. But if you decided that you wanted to wait to encode the TIME HAS FINALLY COME!

3.6.1 The Left Fork - No Encoding, Just Converting Data Back After `missForest`

Ultimately, we need to turn the factor features back into numbers for our next analyses but we had to turn them into factors for missForest. So, now we need to turn them back. Notice a bit of annoying trickery here. Because we had converted them to a factor type, in order to back-convert them correctly to our original ordinal encoding, we had to first convert to character and then to numeric. So, our conversion goes factor $\rightarrow$ character $\rightarrow$ numeric in order to get exactly what we need! Again, we must remember to perform on BOTH imputedTrain and imputedTest.

## Uncomment if this is your choice!
# readyTrain <- imputedTrain %>%
#   mutate_if(is.factor, as.character) %>% 
#   mutate_if(is.character, as.numeric)
# 
# readyTest <- imputedTest %>%
#   mutate_if(is.factor, as.character) %>% 
#   mutate_if(is.character, as.numeric)

3.6.2 The Right Fork - Frequency Encoding

The time has FINALLY come to do frequency encoding, if that is the fork in the road you took all the way back in Section 1.3. Once again, we will be using a helper function that we load with the source() function like we did here. It will do the frequency encoding for us on the State variable, if it needs to be done. If you started from the fully encoded dataset, there is a conditional statement that will skip this for you.

## This calls the helper function in the associated .R file
source(file = "doFrequencyEncoding.R")

## If State is not already a number
if(!is.numeric(imputedTrain$State)) {
    ## This sets a list of COLUMN NAMES I want to frequency encode
    cols2encode <- c("State")
  
    ## This runs the function, which takes up to 4 arguments
    ## Open the source code to see what each argument does!
    ## It returns a LIST of frequency encoded dataframes.
    freqEncoded <- doFrequencyEncoding(train = imputedTrain,
                                       test = imputedTest,
                                       cols2encode = cols2encode, 
                                       quiet = FALSE)
    
    ## Lastly, extract the newly encoded, imputed, dataframes!
    readyTrain <- freqEncoded$train
    readyTest <- freqEncoded$test
}

Question 18: [1 point]

Because we are now frequency encoding AFTER data partitioning (we did ordinal encoding BEFORE the split for convenience), we absolutely must apply the same encoding map to the test set that we did to the train set. Explain why, and what should we do if some classes of the variable end up in the train that is not in the test or vice versa? (HINT: Look at the source code!)

3.7 Transformation & Scaling

Ultimately, we know we will need to scale & center our data, as that is required for the analyses I have set out in our Analysis Plan. But before we get into that, it would be nice to know what realm of analyses in which we fall and, more importantly, do we need to consider transformation of our target in addition to the scaling & centering we plan to do on the entire dataset.

In other words: is our target variable approximately normally distributed?

Figure 7. Histogram of Raw Predicted Readmission Rate

Question 19A: [1 point]

Why is a histogram, or even a Q-Q plot, an insufficient way to assess whether a distribution is approximately normal?

Now Apply a Shapiro-Wilk test for normality using the shapiro.test() function. What does it indicate about your distribution? (Note: Shapiro tests are only reliable for $N < 5000$, but it is fine to perform here.)

# Your code here.

3.7.1 Box-Cox Transformation

The Box-Cox transformation is a family of power transformations used to stabilize variance and make data more normally distributed (Box & Cox, 1964). Recall from our brief discussion in lecture that the Box-Cox transformation involves the parameter $\lambda$, which when applied to $y$ yields the transformation. It is defined as

$y^\lambda = \frac{y^\lambda - 1}{\lambda}$ for $\lambda \ne 0$.

When $\lambda = 0$, it is the same as the $\ln(y)$. Our lecture slides have more reference values for how to interpret $\lambda$. But if $\lambda \ne 0$, then this means that the transformation is only applicable to positive values. The end result is that it helps improve the performance of statistical models that assume normality.

3.7.1.1 There are multiple ways to apply the Box-Cox transform in R, but we are going to take advantage of the one in `caret` to make our lives easier, `BoxCoxTrans()`.

bc_data <- BoxCoxTrans(readyTrain$PredictedReadmissionRate)
print(paste0("Box-Cox lambda =  ", bc_data$lambda))

## [1] "Box-Cox lambda =  -0.3"

We can see that the estimated $\lambda = -0.3$, so approximately a square-root transform (see lecture slides for reference). I found that $\lambda = -0.5$ in FY 2025, for those using that year’s data.

3.7.1.2 Now, apply the Box-Cox transformation to the `train` data:

readyTrain$bc_PredictedReadmissionRate <- predict(bc_data, readyTrain$PredictedReadmissionRate)

3.7.1.3 Assess how it did:

Figures 8a-b. Predicted Readmission Rate After Box-Cox Transformation

Can you see what it did to the distribution? look closely, if not. It’s subtle but discernible!!

Question 19B: [1.5 points]

Now repeat all of the steps to perform a Box-Cox transform on your own to the imputedTest data. Why must we (1) also transform the test data and (2) use the same $\lambda$ as the training data?

# Your code here.

# Make sure to check that your transform looks successful!

NOTE At this point, you have both an non-transformed (PredictedReadmissionRate) and Box-Cox transformed (bc_PredictedReadmissionRate) version of the target in your data.

3.7.2 Centering & Scaling

Now that we have applied a Box-Cox transform to the target variable, we can center and scale the remainder of the variables. We had to perform the Box-Cox prior to centering because the data has to be positive for a Box-Cox transformation (unless we switch to a Yeo-Johnson transform (Yeo & Johnson, 2000)). Centering will likely make a lot of the data negative, plus we know we have negatives in our dataset purposefully.

It’s time to drop the original target, PredictedReadmissionRate, and center and scale everything INCLUDING the target! Here it is for the training set:

readyTrain <- readyTrain %>% 
  ## Drop the original target variable
  select(-PredictedReadmissionRate) %>% 
  ## Make anything that might still be a factor from encoding is back to number
  mutate_if(is.factor, as.numeric) %>%
  ## Center and scale everything!
  scale(center = TRUE, scale = TRUE) %>% 
  data.frame()

Question 19C: [0.5 points]

Do the same for your test data. Make sure to use readyTest and overwrite it.

## Your code here.

Hint: Make sure to turn your centered, scaled matrix back into a dataframe! By default, it creates a matrix.

3.7.3 To save time, you are welcome to start from here after you’ve worked through the above steps.

load(file = "FY2024_data_files/readyTrain.Rdata")

load(file = "FY2024_data_files/readyTest.Rdata")

4 Assessing Multicollinearity & Feature Redundancy

As we finally, finally wrap-up pre-processing there is one final assessment we need to perform: multicollinearity. As I am certain you recall, multicollinearity is when we have high correlation between predictors (so much so, in fact, that it comes at the cost of any correlation with the target!). Recall from lecture that multicollinearity is when one or more of our predictors are moderately to highly correlated with each other. This can inflate our coefficients and thereby cause spurious associations through false positives (i.e., small p-values when we don’t actually have them!).

Here we will implement two common ways we assess multicollinearity: pairwise correlations and Variance Inflation Factors (VIFs).

4.1 Pairwise Correlation

Pairwise correlation is usually done as a Pearson or Spearman correlation. We will calculate a correlation matrix and look at the correlation of the variables within the training dataset to assess possible areas of multicollinearity. We can then visualize this matrix with a heatmap, similar to how we did with missingness, except now blue indicates a positive correlation, red indicates a negative correlation, and yellow indicates no (or zero) correlation. Darker colors indicate stronger correlations.

Figure 9. Correlation Heatmap of All Variables

Question 20A: [1 point]

Where do you see the highest potential for multicollinearity? Why do you say that?

Which variable(s) seem to be most correlated with our target, bc_PredictedReadmissionRate?

4.2 Fitting a model to estimate VIF

Another way we can assess multicollinearity is by fitting a multiple linear regression model and estimating the variance inflation factors, or VIF, which estimates how much the variance of an estimated regression coefficient increases if your predictors are correlated. Specifically, variance inflation can identify multicollinearity when the VIF is greater than ~4 and especially when it is above 10.

4.2.1 Fit a linear regression with the training data, using the Box-Cox transformed `Predicted Readmission Rate` as the target.

mod <- lm(bc_PredictedReadmissionRate ~ ., data = readyTrain)

4.2.1 Next, let’s calculate the VIFs using the `vif()` function that is part of the `car` package.

Note that I am not printing all of the VIFs, just the top 15 after sorting in descending order.

## Calculate the VIFs and put into a dataframe to print the table
vifResults <- vif(mod) %>% data.frame
## Change the column names of the dataframe
colnames(vifResults) <- c("VIF")

Table 7. Top 15 Variance Inflation Factors after multiple linear regression
	VIF
Overall.hospital.rating	22.69
Recommend.hospital	14.77
Nurse.communication	8.35
Care.transition	8.02
Staff.responsiveness	5.26
Communication.about.medicines	4.59
Doctor.communication	4.19
Discharge.information	3.48
Composite.patient.safety	2.74
Quietness	2.35
ComparedToNational_Composite.patient.safety	1.87
Death.rate.for.pneumonia.patients	1.85
Cleanliness	1.79
ComparedToNational_Death.rate.for.pneumonia.patients	1.76
Postoperative.respiratory.failure.rate	1.56

Question 20B: [1 point]

Does multicollinearity seem to be an issue in this dataset and, if so, among which variables? Why do you come to this conclusion?

HINT 1: What was used to calculate Overall Hospital Rating? Does this help us better understand why the VIF is so large?

HINT 2: Is, for example,Payment Category for pneumonia patients created from Payment for pneumonia patients? [These are Medicare payments]? Which one do you think we should keep amd why?

Which variable(s) seem to be most highly correlated with our target? Is it a positive or negative correlation? Does it make sense?

4.3 Dropping the most collinear variables

Even though we plan to proceed with an elastic net (which is much more robust to multicollinearity), some of these VIFs are too high for comfort. We also want to pause and ask - were any of the variables used to make other variables? (The answer is “yes”!). Those are excellent candidates to scrub too; we want to use our domain knowledge and what our stakeholders are asking for to help us decide which ones to keep.

Because the goal is a deployable predictive model, we likely want something more granular than most of the ComparedToNational features can give us, with the exception of our key predictor, ComparedToNational_Hospital return days for pneumonia patients. I will remind you of what we explored in Section 2 with this variable; not only that, we can see how how highly correlated it is with our target!.

4.3.1 Drop variables with VIF over 5 from both the training and testing data

## Drop those with VIF over 5
vifOver5 <- vifResults %>% filter(VIF >=5) %>% rownames()
print(vifOver5)

## [1] "Nurse.communication"     "Staff.responsiveness"   
## [3] "Care.transition"         "Overall.hospital.rating"
## [5] "Recommend.hospital"

## Now drop them
readyTrain <- readyTrain %>% select(-any_of(vifOver5))
readyTest <- readyTest %>% select(-any_of(vifOver5))

4.3.2 Now drop all of the `ComparedToNational` and `PaymentCategory` features except `ComparedToNational_Hospital return days for pneumonia patients`

readyTrain <- readyTrain %>% 
  ## Rename the column we want to keep
  rename(Hospital.return.days.for.pneumonia.patients = ComparedToNational_Hospital.return.days.for.pneumonia.patients) %>% 
  ## Drop all the other columns 
  select(-contains("Compared"), -contains("PaymentCategory"), -SurveyResponseRate)

readyTest <- readyTest %>% 
  ## Rename the column we want to keep
  rename(Hospital.return.days.for.pneumonia.patients = ComparedToNational_Hospital.return.days.for.pneumonia.patients) %>% 
  ## Drop all the other columns 
  select(-contains("Compared"))

5 Unsupervised Learning Methods: Segmentation Analysis

Predictive modeling is our ultimate goal, but let’s pretend our stakeholders have expressed a desire to better understand how the hospitals relate to one another and whether there are any general patterns we should be interested in for enhanced interpretation. After all, a prediction is one thing, but understanding the why is still usually important!

Thus, we have opted to undertake a Segmentation Analysis to see if there is any underlying segments, or clusters, of hospitals based on the data we have. We will use the unsupervised methods of Principal Component Analysis (PCA) and $k$-means clustering do this; it will help us to determine if there are broader classifications of our hospitals that we need to report back to the advocacy group.

5.1 Principal Component Analysis

First, we apply a PCA on the data and display the principal components. Note that we have the same number of principal components as we do variables in the data set! The cumulative proportion of variance adds to 100% by our last principal component; but we’re really more interested in the portion of variance explained by EACH of the principal components. This can help us figure out just how many PCs are important for explaining the most variance. For example, if we were interested in feature reduction, we could use the top PCs that explain, say, some portion of the variance that we wish to retain.

Question 21: [2 points]

At what principal component do we surpass 80% of explained variance? (HINT: look at the cumulative variance row rather than tha proportion of variance row).

pca <- prcomp(readyTrain)
summary(pca)

## Importance of components:
##                           PC1    PC2     PC3     PC4    PC5     PC6     PC7
## Standard deviation     1.9670 1.3511 1.23353 1.12735 1.1017 0.96189 0.95477
## Proportion of Variance 0.2276 0.1074 0.08951 0.07476 0.0714 0.05443 0.05362
## Cumulative Proportion  0.2276 0.3350 0.42447 0.49923 0.5706 0.62505 0.67867
##                            PC8     PC9    PC10    PC11    PC12   PC13    PC14
## Standard deviation     0.92789 0.89813 0.85670 0.82968 0.76062 0.7260 0.68587
## Proportion of Variance 0.05065 0.04745 0.04317 0.04049 0.03403 0.0310 0.02767
## Cumulative Proportion  0.72932 0.77677 0.81994 0.86043 0.89446 0.9255 0.95314
##                           PC15    PC16    PC17
## Standard deviation     0.59103 0.50709 0.43608
## Proportion of Variance 0.02055 0.01513 0.01119
## Cumulative Proportion  0.97369 0.98881 1.00000

You may also notice that this is always ordered from most proportion of variance on the left (thus is PC1 defined) to our final principal component on the right. What percent of variance is explained by just PC1? What percent by the final PC? Is the attributable portion of the variance in PC1 high, medium, or low based on some quick research or prior knowledge?

We use scree plots to graphically determine how many principal components to retain and how many principal components best explain the variance of the data. The $x$-axis displays the principal components (PC1, PC2, etc.), while the $y$-axis shows the eigenvalues (or proportion of variance explained) for each component. The elbow method we discussed in class is the point where the slope starts to flatten or the point at which the percentage of variance explained seems to level off. If we were doing this for feature reduction, we might think of this as approximately how many principal components would be needed to explain the total variation in the training data. However, it is a heuristic method, so it can sometimes be a little uncertain from the plot alone what the optimal number of PCs is. Let’s take a look at the top-15 PCs on a scree plot now:

Figure 10. Scree Plot of First 15 Principal Components

How many PCs appear optimal to you based on the scree plot?

People often mis-write scree (I’ve even accidentally done it) as SCREE, as if it’s an acronym like AUROC or ROC or even PCA. But it’s not. Why is it called a scree plot? (Hint: It may not be what you expect!)

5.1.2 Variable loadings and contributions

Next, let’s explore how much the variables are contributing (this is based on the weight of their rotations). In PCA, we are interested in the loadings, i.e., the linear combination weights (coefficients) whereby unit-scaled components define or “load” a variable. Loadings help us interpret principal components.

You may have noticed we used the prcomp() function above; prcomp() will return the loadings in the variable $rotation, which contains a matrix of variable loadings. These loadings have been determined from the eigenvectors, which without getting into what that is (because we’d have to take a departure into linear algebra), suffice it to say that they are how we are measuring the direction and magnitude during each subsequent rotation of the principal components as we measure the variance explained.

Additionally, we are going to color this by our grouping variable and key predictor in the dataset, Hospital.return.days.for.pneumonia.patients. Recall that when we explored this earlier this variable is the performance of each hospital in terms of its pneumonia-related return-to-hospital days relative to the national average. In other words, hospitals that were “Worse than average” had readmitted pneumonia patients with longer-than-average stays.

Figure 11. PCA Biplot Showing the Top Feature that Contributes to Explained Variance

This biplot (Figure above) shows that, on principal components 1 and 2 (the PCs that account for the most variance in the training data) there is subtle but noticeable grouping of hospitals in terms of how long their readmitted pneumonia patients stayed in the hospital. I have chosen to display the #1 feature that contributes to that variance, Communication.about.medicines.

Each quadrant on a PCA biplot corresponds to a different directional influence of the original variables. E.g., because it is in the top-right quadrant I where both PC1 and PC2 are positive, this suggests that communication about medicines are positively associated with variables that point into the same quadrant. Since it seems to correspond with where the “better” hospitals are, this suggests that these hospitals are made somewhat distinctly “better” at readmitted hospital stay lengths because of correlations with communication about medicines. (NOTE: If you look at the 2025 data, the bipolot is effectively inverted but the conclusions seem to be the same in 2025 as well).

A general rule of thumb is:

Top-right Quadrant I

PC1 and PC2 are both positive
Thus points in this quadrant are positively associated with variables pointing into this quadrant
Indicates high values for variables with arrows in this quadrant

Top-left Quadrant II

PC1 is negative, PC2 is positive
Thus points here are associated with high values of variables pointing to the left and top
These observations contrast with those in the right-side quadrants on PC1

Bottom-left Quadrant III

PC1 and PC2 are both negative
Thus points here are negatively associated with variables pointing into the other quadrants
These points could represent a distinct subgroup or pattern with behavior opposite to other points.

Bottom-right Quadrant IV

PC1 is positive, PC2 is negative
Thus points here are associated with variables pointing into this quadrant
May indicate a contrast with Quadrant II on PC2

Question 22: [1 point]

Play around with the graphical parameters in the biplot, especially those I’ve marked with comments. Try adding, for example, more variables to display rather than just the top contributor. NOTE that if you find variables in the same quadrant together, this means that they are correlated. What variable(s) can you find that contrast with Communication.about.medicines?

Now, look at the contributions to PCs 1 and 2 (Figure Below) from the variables in the training dataset. Anything above the dotted red-line contributes significantly to the overall explanation of variance, and anything below the dotted line does not. Does anything about this plot stand out to you?

HINT 1: What do the top contributing variables all have in common? In other words, did they all come from the same data source?

HINT 2: Where does the target, PredictedReasmissionRate, fall among the significant variables?

Figure 12. Variables that Significantly Explain Variance.

5.2 k-means Clustering

$k$-means is a centroid-based clustering algorithm, where a “centroid” is the geometric center of an object often measured by Euclidean distance. In the algorithm, the distance between each data point and a centroid is calculated by randomly grabbing data; these distances are then used to assign the data point to a cluster. The goal is to identify the optimal $k$ number of groups in a dataset by minimizing the distance of each datapoint to a respective centroid.

5.2.1 Visualizing Clusters

Although multiple methods exist to determine the optimal number of clusters, one commonly employed is heuristic, i.e., it isn’t really computational. You will either choose the optimal clusters based on a set of options of $k$ graphically or you can once again employ the “elbow” method again, as you did with PCA. Let’s start by visually inspecting the effect of different $k$ clusters on PC1 and PC2 of the dataset. I have chosen to test $k$ groups of 2 through 9.

Figure 13. Biplots showing possible k clusters

Question 23A: [0.5 points]

Which number of clusters, $k$, do you think has the best explanatory power? Why? Is it is hard to tell?

5.2.3 Within-sum-of-squares (WSS) elbow method

Next, let’s employ the ‘elbow’ method again, this time to look at when the within sum of squares drops off rather than the percent variance, as was done in PCA.

Figure 14. WSS Scree plot for k-means

Question 23B: [1 point]

What is the within sum of squares and what exactly is it measuring here? By the elbow method, which $k$ is the optimal number of clusters? Did it agree with your choice from the other heuristic method?

5.2.3 Silhouette method

Lastly, we will apply the silhouette method, which measures the average silhouette width, a combination of (1) how similar any given point is to its own cluster (also called cohesion) and (2) how different that same point is from other clusters (also called separation). Silhouette scores can range from $-1 \rightarrow +1$, with larger values indicating better clustering (i.e., more cohesive clusters that are separated from each other). Thus, a silhouette score of 1 indicates perfect separation and cohesion whereas a silhouette of 0 indicates overlapping clusters.

What we can do is iteratively test our possible $k$ values of 2 to 9 and a silhouette score will be calculated for each $k$ tested. The highest silhouette score indicates the optimal $k$.

Figure 14. Silhouette Plot for k-means

Question 23C: [0.5 points]

Are you surprised at the optimal $k$ from the silhouette method? Does it seem to agree with what you’ve seen with the WSS elbow method? (Hint: agreement generally suggests evidence that you have found the optimal $k$).

Lastly, let’s extract the optimal $k$ from the silhouette method.

## Extract optimal k from silhouette method; first get the cluster data
clusterData <- fviz_nbclust(readyTrain, kmeans, method = "silhouette")$data
## Then extract the maximum y
k <- as.numeric(clusterData$clusters[which.max(clusterData$y)])

5.3 Segmentation Analysis

As we said before, the goal of segmentation analysis is to broadly cluster our hospitals by their features to help us characterize them.

Question 24: [1 point]

Let’s explore the clusters - segments - of the hospitals based on their average values from the original dataset (we start from imputedTrain, which hadn’t yet been transformed, centered, or scaled, BUT the multicollinear variables also haven’t been dropped either). Your task is to add comments to this code chunk.

## COMMENT HERE
kmeansResult <- kmeans(readyTrain,  
                         centers = k,      ## COMMENT HERE
                         nstart = 50,      
                         iter.max = 1000,  
                         algorithm = "MacQueen")
## COMMENT HERE
kmeansTrain <- imputedTrain %>% 
  ## Rename the column we want to keep
  rename(Pneumonia.Hospital.Return.Days = `ComparedToNational_Hospital return days for pneumonia patients`) %>% 
  ## COMMENT HERE
  select(-any_of(vifOver5), -contains("Compared"), -contains("PaymentCategory"), -SurveyResponseRate) %>% 
  ## COMMENT HERE 
  mutate(Cluster = kmeansResult$cluster,
         Pneumonia.Hospital.Return.Days = as.numeric(if_else(Pneumonia.Hospital.Return.Days == "1", 1,
                                                     if_else(Pneumonia.Hospital.Return.Days == "0", 0, -1)))) %>% 
  ## COMMENT HERE
  clean_names()

## Using % contribution graph, the order of the top 10:
pcaImportantVars <- factor(c("Communication.about.medicines",
                       "Doctor.communication", 
                       "Composite.patient.safety",
                       "Discharge.information",
                       "Quietness",
                       "Postoperative.respiratory.failure.rate",
                       "Cleanliness",
                       "Periop.Rate.Of.Embolism.Or.Thrombosis",
                       "Predicted.Readmission.Rate",
                       "Pneumonia.Hospital.Return.Days"))

## COMMENT HERE
pcaImportantVars <- tolower(pcaImportantVars)
## COMMENT HERE
pcaImportantVars <- gsub("\\.", "_", pcaImportantVars) %>% 
## COMMENT HERE
  to_any_case("title")

## COMMENT HERE
kmeansTrain %>% 
  ## COMMENT HERE
  rename(Predicted.Readmission.Rate = predicted_readmission_rate,
         Periop.Rate.Of.Embolism.Or.Thrombosis = perioperative_pulmonary_embolism_or_deep_vein_thrombosis_rate) %>% 
  ## COMMENT HERE
  clean_names(case = "title") %>% 
  ## COMMENT HERE
  select(Cluster, any_of(pcaImportantVars)) %>% 
  ## COMMENT HERE
  pivot_longer(-1, names_to = "Measure", values_to = "Value") %>% 
  ## COMMENT HERE
  # filter(Measure %in% pcaImportantVars) %>% 
   mutate(Measure = factor(Measure, levels = to_any_case(c("Communication.about.medicines",
                        "Doctor.communication", 
                        "Composite.patient.safety",
                        "Discharge.information",
                        "Quietness",
                        "Postoperative.respiratory.failure.rate",
                        "Cleanliness",
                        "Periop.Rate.Of.Embolism.Or.Thrombosis",
                        "Predicted.Readmission.Rate",
                        "Pneumonia.Hospital.Return.Days"), case="title"))) %>% 
  ## COMMENT HERE
    ggplot(aes(x = Cluster, y = Value, fill = as.factor(Cluster))) +
    geom_boxplot(alpha=0.75) +
    scale_fill_manual(values = c(skittles[1], skittles[6])) +
    labs(title = "Cluster Means",
         x = "",
         fill = "Cluster") +
    theme_classic() +
    theme(legend.position = "bottom",
        strip.text = element_text(size=8),
        legend.text = element_text(size = 8),
        legend.title = element_text(size = 10),
        title = element_text(size = 12, color = "maroon")) + 
    ## COMMENT HERE
    facet_wrap(~ Measure, 
             scales = "free_y",
             ncol = 2)

Figure 16. Boxplot Comparisons of Important Variables by Cluster

## COMMENT HERE
kmeansTrain %>% 
  ## COMMENT HERE
  rename(Predicted.Readmission.Rate = predicted_readmission_rate,
         Periop.Rate.Of.Embolism.Or.Thrombosis = perioperative_pulmonary_embolism_or_deep_vein_thrombosis_rate) %>% 
  ## COMMENT HERE
  clean_names(case = "title") %>% 
  ## COMMENT HERE
  select(Cluster, any_of(pcaImportantVars)) %>% 
  ## COMMENT HERE
  tbl_summary(by = "Cluster",
              statistic = list(all_continuous() ~ c("{mean} ({median})"))) %>% 
  modify_header(label ~ "Variable",
                all_stat_cols() ~ "**Cluster {level}**") %>% 
  modify_caption("Table 8. Summary Statistics of Important Cluster Variables")

Table 8. Summary Statistics of Important Cluster Variables
Variable	Cluster 1¹	Cluster 2¹
Communication About Medicines	76.5 (76.0)	70.9 (71.0)
Doctor Communication	91.03 (91.00)	87.68 (88.00)
Composite Patient Safety	0.98 (0.96)	1.06 (1.01)
Discharge Information	87.0 (87.0)	82.3 (83.0)
Quietness	83.2 (83.4)	77.1 (77.0)
Postoperative Respiratory Failure Rate	8.69 (8.53)	9.93 (9.25)
Cleanliness	86.4 (86.8)	82.1 (82.6)
Periop Rate of Embolism or Thrombosis	3.55 (3.52)	3.73 (3.59)
Predicted Readmission Rate	15.77 (15.67)	17.57 (17.38)
Pneumonia Hospital Return Days
-1	188 (14%)	430 (50%)
0	976 (72%)	408 (47%)
1	185 (14%)	23 (2.7%)
¹ Mean (Median); n (%)

HINT: If these are back to the original values, this is more interpretable for our stakeholder. We will now read this just like a typical boxplot; for example, median (the middle line) Doctor Communication seems higher in Cluster 1 than it does in Cluster 2 with only some overlap in the distributions (the boxes and whiskers don’t really overlap each other). The table confirms this; the median Doctor Communication score is 91% in Cluster 1 but only 88% in Cluster 2.

Question 25: [2 points]

Notice, for example, that Cluster 1 hospitals seem to include the hospitals that had the lowest pneumonia-related readmission rates, with the average rate ~2% higher in Cluster 2 hospitals. Thus, for our stakeholder, we might decide to rename Cluster 1 as “Highest Performing Hospitals” or something along those lines. Notice also that this cluster had more below national average return hospital days (a “1”). This means that they had more readmitted pneumonia patients who spent less time in the hospital upon readmission than the national average.

Look at how else you might broadly characterize - or segment - these hospitals. Your job is to make a table for your client, summarizing each of the two clusters with FOUR primary criteria. Make sure to give each cluster an informative name as I did; you’re welcome to rename Cluster 1 if you think of a catchier name than I chose!

After you come up with names of the two clusters, you will then come up with at least FOUR characteristics per cluster summarizing what you see in either the boxplots or table. You may choose to focus on different attributes for different clusters. Fill in the code for the table below. I start you out with two examples for Cluster 1, but feel free to add more!

Table 9. Hospital segmentation analysis: three types of broad groupings identified.
Top Defining Attributes	Description
1: Highest Performing Hospitals
Predicted Pneumonia-related hospital readmissions	These hospitals have a lower readmission rate, suggesting that hospitals in this cluster may be better at diagnosing and treating pneumonia
Return hospital days due to pneumonia	Readmitted patients spend below the national average number of days in hospital, suggesting these hospitals are able to more quickly treat and stabilize pnuemonia patients
FILL ME IN	FILL ME IN
FILL ME IN	FILL ME IN
2: NAME ME!
FILL ME IN	FILL ME IN
FILL ME IN	FILL ME IN
FILL ME IN	FILL ME IN
FILL ME IN	FILL ME IN

5.4 Final Remarks on Segmentation

We don’t technically have to stop our segmentation analysis here, but we will so that we can focus on other analyses. One thing we might choose to do in the future, for example, is to choose a classification machine learning algorithm to test the predictions based on these clusters.

Question 26: [1 point]

Make a recommendation for our client for 1-2 machine learning analyses your team would choose to use to test whether these clusters can be used for robust predictions of hospital performance for pneumonia patients. How would you know whether or not your predictions were robust? What would you look for or compare?

6 Supervised Learning Methods

We are going to perform two supervised learning methods: an Ordinary Least Squares (OLS) regression followed by an Elastic Net.

6.1 Ordinary Least Squares (OLS) regression

Earlier in Section 4 when we assessed multicollinearity, we fit an OLS regression model so that we could calculate variance inflation factors (VIFs). However, because OLS regression is so incredibly sensitive to multicollinearity, we should re-run it now that we’ve removed some of the most redundant variables.

Let’s model that now, using the readyTrain dataset and plot some diagnostic plots. to assess the key assumptions of an OLS linear regression. The four plots are diagnostic plots for the primary assumptions of a linear regression:

Linearity between the outcome and the predictors
Normally distributed residuals
Homosecadasticity (equality of variances) in the residuals
No high leverage/influence points (no significant outliers)

Figure 17. OLS Regression Assumption Diagnostic Plots

Question 27A: [1 point]

What is your general assessment of our OLS assumptions? Met or unmet? Why?

Note: If you are not entirely sure about the Q-Q plot, you can run a Shapiro-Wilk test on the residuals using the following code:

shapiro.test(residuals(mod))

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(mod)
## W = 0.99862, p-value = 0.06735

The most important OLS assumption for Elastic Net is linearity of the relationship between predictors and the response. The other, stricter OLS assumptions don’t matter as much because of regularization.

Do you find support for linearity here? Why or why not?

Let’s now assess how well our model fits with the adjusted $R^2$.

## [1] 0.39427

Question 27B: [1 point]

It is a surprisingly high (although it isn’t stellar). You may need to do some outside research, but explain in a sentence or two why, generally speaking, we use adjusted $R^2$ and not the unadjusted one (confusingly labeled as multiple r-squared in the summary() output below).

Find the best model.

Use the step() function to do a backward regression to find our most parsimonious model. Stepwise regression drops terms that are not significant or important, thereby only keeping the features that contribute to the overall explanation of variance in predicted readmission rate.

## Perform a backward, stepwise regression to find the most parsimonious model
bestMod <- step(mod, 
            direction = "backward", 
            trace = FALSE)

Lastly, let’s take a peek at the results. We will use a package called stargazer to help us visualize the results in a nicely organized way.

## Print a table using stargazer
stargazer(bestMod, 
          type = "html",
          title = "Table 10. Parsimonious OLS Regression Results.")

**Table 10. Parsimonious OLS Regression Results.**

	Dependent variable:

	bc_PredictedReadmissionRate

MRSA.Bacteremia	0.039^**
	(0.017)

Abdomen.CT.Use.of.Contrast.Material	0.045^***
	(0.017)

Death.rate.for.pneumonia.patients	-0.154^***
	(0.017)

Medicare.spending.per.patient	0.167^***
	(0.018)

Healthcare.workers.given.influenza.vaccination	0.044^**
	(0.018)

Left.before.being.seen	0.062^***
	(0.017)

Venous.Thromboembolism.Prophylaxis	0.101^***
	(0.018)

Doctor.communication	-0.074^***
	(0.027)

Discharge.information	-0.053^**
	(0.024)

Cleanliness	-0.053^***
	(0.020)

Quietness	-0.077^***
	(0.022)

Hospital.return.days.for.pneumonia.patients	-0.396^***
	(0.018)

State	0.110^***
	(0.019)

Constant	0.000
	(0.017)


Observations	2,210
R²	0.398
Adjusted R²	0.395
Residual Std. Error	0.778 (df = 2196)
F Statistic	111.824^*** (df = 13; 2196)

Note:	p<0.1; p<0.05; p<0.01

Lastly, let’s assess how good a predictive model our OLS regression model is. We will use the predict() function to first fit our most parsimonious model bestMod to the test data:

## Make the predictions
predictions <- predict(bestMod, newdata = readyTest)

The Root Square Mean Error ($RMSE$) is the square root of the average of the squared differences between predicted and actual value, such that $RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^n\cdot(y_i-\hat{y}_i)^2}$ where $y_i$ is each observed value, $\hat{y}_i$ is each corresponding predicted value, and $n$ is the number of observations. NOTE: Recall that a residual is $y_i-\hat{y}_i$ - the distance between every point and its predicted value! It’s also important to realize that the $RMSE$ gives more weight to large errors because of squaring. Thus, it’s sensitive to outliers.

Question 27C: [1 point]

Can you interpret what the RMSE tells your client here? Why or why not? What about the $R^2$?

Hint 1: Remember that the outcome here is Box-Cox transformed. Would you need to undo the transformation to be able to assess what the RMSE (or $R^2$) is actually telling us? Recall that the Box-Cox $\hat{y} = \frac{1}{y^\lambda}$ where our $\lambda = -0.3$ AND we also centered and scaled these data.

Hint 2: Is it practical to undo both transformations? How does that hinder our ability to make an interpretation for our client?

Lastly, let’s graph the effect predictive ability of our OLS model by comparing the relationship between the true values of bc_PredictedReadmissionRate from the readyTest dataset to the predictions we made using the model on the testing data.

Figure 18. Predictions Plot for OLS Regression

NOTE that, while there is an obvious linear trend, it is weak. There is a lot of error around the predictions from the OLS regression!

6.2 The Elastic Net

Perhaps the seemingly low $R^2$ is because we still have too much multicollinearity or overfitting. So, as we discussed in lecture, we can address this using regularized regressions, such as the Elastic Net, which is more robust to multicollinearity by assessing penalties for non-zero variance. Typically, we would likely conduct a Ridge and/or LASSO before running an Elastic Net, but since we already know that (1) we have a fairly high amount of multicollinearity in our original readyTrain dataset and (2) we know we want to be able to perform feature reduction and figure out which features are important predictors of hospital readmissions, we’re going to jump straight into Elastic Net, which performs both.

6.2.1 The Elastic Net Unpacked

In general, regularization is a technique that applies a penalty term to a cost function of a machine learning model, like the OLS regression. The reason is to discourage overfitting. This penalty term constrains the model’s coefficients, which limits how flexible they are during training. Why?

The more flexible the coefficients, the less likely they will “learn” new data, i.e., the more generalizable they are! Thus, by applying this penalty (constraint), it improves the model’s performance on “unseen” or new data.

As mentioned in lecture, Elastic Net regression is a general regularization technique that combines the regularization techniques of Ridge and LASSO to help us perform both feature selection (i.e., significant coefficients) and feature reduction (importance). The elastic net can even help us with the $P >> n$ problem! Elastic net has two regularization terms, called L1 (or the Lasso regularization term) and L2 (the Ridge term).

L1: makes some of the coefficients zero, thereby selecting only the most important features (shrinks them to zero). It is often said that this encourages sparsity in the model.
L2: reduces the coefficients to small but non-zero values, thereby reducing the coefficients of the unimportant features (which we can use to determine significance). It does this by adding a penalty term to the cost function of the model, which is proportional to the coefficient squared. This allows us to retain all the features in the model but reduces the overall impact of the non-significant or less-important ones.

Thus, by combining these techniques together, the Elastic Net is balancing feature selection with feature reduction! If you’re interested in reading the original paper on Elastic Net by Zou and Hastie (2005), you can find a free PDF of the paper here.

6.2.2 Running, tuning, & cross-validating the Elastic Net

The elastic net can be run using the glmnet package (which we are accessing via caret), and can be cross-validated as random forest and SVM from our last project could be. This is a big improvement over OLS regression, which cannot be tuned. Additionally, the elastic net has two hyperparameters that we can tune, $\alpha$ and $\lambda$ (not to be confused with the $\lambda$ from Box-Cox transformation - Greek letters are abundant in statistics! Why do you think I named our course teams as I did?!).

$\alpha$ is the strength of the regularization, representing the balance between L1 and L2 regularization. Larger $\alpha$ results in L1 regularization (feature selection / Ridge) whereas smaller $\alpha$ results in more L2 regularization (higher shrinkage / feature reduction / Lasso). When $\alpha=0$ the regression is equivalent to OLS regression! $\alpha$ is also referred to as the mixing percent of L1 and L2 regularization.
$\lambda$ is the shrinkage parameter, such that when $\lambda = 0$ no shrinkage is performed and is equivalent to an OLS regression. As $\lambda$ increases, the coefficients are increasingly shrunk and thus the more features will have coefficients shrunk to zero, regardless of the value of $\alpha$.

6.2.2.1 We will start by setting our CV parameters just as we did on the last project, using the `trainControl()` function in `caret`.

NOTE that I am performing a 5-fold repeated cross-validation (CV) with 5 iterations.

ctrl <- trainControl(method = "repeatedcv",  ## Do repeated CV
                     number = 5,             ## Number of k-folds
                     repeats = 5,            ## Number of repeats
                     search = "grid",        ## Grid search (vs. random)
                     verboseIter = FALSE)    ## Don't show me all the results

6.2.2.2 Then, we fit the model using the `train()` function in `caret`, specifying that we want to perform an elastic net, which will tune the model and perform CV per the specifications in the `trainControl()`. NOTE that I am allowing the values of $\alpha$ to range from $0 \rightarrow 1$ and allowing $\lambda$ to range from $0 \rightarrow 5$.

## Create a search grid that allows alpha to range from 0 --> and 
## allows lambda to range from 0 --> 5
searchGrid <- expand.grid(.alpha = seq(0, 1, length.out = 10), 
                          .lambda = seq(0, 5, length.out = 15))

## Fit the elastic net using the tuning grid and CV scheme laid out
elasticMod <- train(bc_PredictedReadmissionRate ~ ., 
                    data = readyTrain, 
                    method = "glmnet", 
                    tuneGrid = searchGrid,
                    trControl = ctrl)

Question 28: [1 point]

Explain what we are doing here. You may want to look up the parameters of the trainControl() and train() functions. Make sure to address what type of CV and hyperparameter search is being performed, as well as if both the $\alpha$ and $\lambda$ hyperparameters of elastic net are being tuned.

Hint 1: Why are we doing cross-validation, and specifically why did I choose the one I did?

Hint 2: What does $\lambda = 0$ mean? How about $\alpha = 0$?

6.2.3 Elastic Net Results

6.2.3.1 We are going to first take a look at the top 15 results of the Elastic Net in a table form, sorted from lowest $RMSE$:

Table 11. Top Performing Results of the Elastic Net Tuning & 5-fold Repeated Cross-Validation
alpha	lambda	RMSE	Rsquared	MAE	RMSESD	RsquaredSD	MAESD
0.00	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.33	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.56	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.44	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.22	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.67	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.78	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.11	0.00	0.78	0.39	0.62	0.02	0.03	0.02
1.00	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.89	0.00	0.78	0.39	0.62	0.02	0.03	0.02
0.00	0.36	0.79	0.39	0.63	0.02	0.03	0.02
0.11	0.36	0.80	0.38	0.63	0.02	0.03	0.02
0.00	0.71	0.80	0.38	0.64	0.02	0.03	0.02
0.00	1.07	0.82	0.37	0.65	0.02	0.03	0.02
0.00	1.43	0.83	0.36	0.66	0.02	0.04	0.02

NOTE: Folds that fail to converge will have an $R^2$ = NA.

6.2.3.2 We can also extract the hyperparameters of our best tuned model:

$\alpha$:

## [1] 0

$\lambda$:

## [1] 0

Question 29: [1 point]

Check the values of $\alpha$ and $\lambda$ here. What do they suggest about the optimal regularization that is being fit here? (Recall that when both $\alpha$ and $\lambda$ are zero, it’s an OLS regression!)

Let’s also plot the results of the tuning search. Can you figure out the type of tuning search you did? (Hint: the visual pattern should confirm your answer from earlier!) The diamond denotes the best combination of hyperparameters!

results <- elasticMod$results %>% 
  data.frame()
best <- results[as.numeric(rownames(elasticMod$bestTune)), ]

Figure 19. Hyperparamater Grid Search for Elastic Net

6.2.3.3 Important Features

Important features can also be extracted, using the varImp() function from caret.

important <- varImp(elasticMod)$importance

Figure 20. Variable Importance from Best Elastic Net Model

Question 30: [1 point]

Which of the features were most important? How does this compare to the most important features out of OLS regression? Make sure you’re looking at the absolute value of the beta coefficients to assess importance in OLS regression.

How about about of PCA? You may have already noticed that importance almost seems inverted! Why?!

Hint 1: What is the fundamental difference between a PCA and any regression?

Hint 2: What does “importance” mean in a PCA vs. any regression?

Hint 3: What does the sign of the loadings mean in a PCA? Are they actual or arbitrary?

6.2.3.2 Check model performance

Assessing model performance means assessing how well our training model does when applied to “unseen” data, here our readyTest dataset. Because we know the right answer - the true value of PredictedReadmissionRate, we can assess the error that the training model produces when fitted against the testing data. We measure that error as $RMSE$ and $MAE$. We briefly reviewed what the $RMSE$ measures back in Question 36, so I will only review what the $MAE$ is here.

Mean Absolute Error ($MAE$) is the average of the absolute differences between predicted and actual values, such that $MAE = \frac{1}{n}\sum_{i=1}^n\cdot|y_i-\hat{y}_i|$ where $y_i$ is each observed value, $\hat{y}_i$ is each corresponding predicted value, and $n$ is the number of observations. So, unlike $RMSE$, $MAE$ treats all errors equally and therefore is more robust to outliers. So, $MAE$ will tell you the average magnitude of the error, regardless of direction!

Let’s also make predictions on the testing sets for both the OLS and Elastic Net models using the postResample() function in caret; this will not only do predictions, but it will enable us to more easily generate metrics that we can use to compare model performance, such as $R^2$ and MAE.

## Post-resample on OLS
prOLS <- postResample(pred = predict(mod, newdata = readyTest), 
                      obs = readyTest$bc_PredictedReadmissionRate)

## Post-resample on Elastic Net
prElastic <- postResample(pred = predict(elasticMod, newdata = readyTest), 
                      obs = readyTest$bc_PredictedReadmissionRate)

Table 12. Comparison of Model Performance of OLS and Elastic Net Regressions
	RMSE	Rsquared	MAE
prOLS	0.81	0.35	0.64
prElastic	0.81	0.35	0.64

6.2.3.3 Check for overfitting/underfitting

Question 31: [1 point]

Using the same method, check for overfitting / underfitting. Make predictions using readyTrain rather than readyTest, and then check how the RMSE on the training data compares to the RMSE of the training data for the Elastic Net. How do they compare? (Recall what we discussed when we did the gene expression demo: the relationship of the test vs. training accuracy can be used to assess overfitting/underfitting. The same logic applies to RMSE too!) How do both compare to what we got out of the OLS regression? Which would you say is the better predictive model, the OLS regression or the Elastic Net? Why?

## Post-resample on OLS
prOLS <- postResample(pred = predict(mod, newdata = readyTrain), 
                      obs = readyTest$bc_PredictedReadmissionRate)

## Post-resample on Elastic Net
prElastic <- postResample(pred = predict(elasticMod, newdata = readyTrain), 
                      obs = readyTest$bc_PredictedReadmissionRate)

Table 13. Comparison of Training Performance of OLS and Elastic Net Regressions
	RMSE	Rsquared	MAE
prOLS	NA	0	NA
prElastic	NA	0	NA

6.2.3.4 Graphical comparison of the two models’ predictions

Figure 19. Predictions Plot for Elastic Net vs. OLS Regression

It should not come as a surprise, but these models are performing VERY similarly!

Question 32: [4 points]

What conclusions do you make at this point about whether to make predictions with an OLS regression, Elastic Net, or neither? What recommendations would you make to our client at this point? What other analyses might your suggest, or other data we might want to include? Please try to flesh out 2-3 recommendations in at least a paragraph.

HINT: Make sure to ask yourself whether these data appear linear or not. Could non-linearity be a cause for a low $R^2$? If not, what other causes are there and how will that impact your recommendations?

7 Choose Your Own Adventure

It’s time to choose your adventure! After working through this demo and answering the 31 questions, you will choose one of three possible trajectories. The following are some brief hints or tips for each of the choices available to you.

Adventure 1.

[Choosing one other condition to analyze.]

Make sure that you use the code I provided, updating it for your new condition
Make sure to answer all of the interpretative questions again for your new dataset.
I will specifically be looking for a comparison of which condition, pneumonia or the one you chose, seems to be a better choice for a predictive model for our client. Or is it neither?
Make sure to include some next steps or recommendations based on your new analysis!

Recommendations & Hints:

You can avoid going back to Demo 1 to do your importing, cleaning, and pre-processing by running this source code:

## Make any necessary changes in the R source code
## Uncomment to run!
# source(file = "reRunDemoData.R")

Make sure to update your filepath to where you stored the hospital files on YOUR local machine, just as you did in Question 1 of the Demo!
Make sure to update the condition that gets selected from “PN” to the condition of your choosing. A full list of available conditions is found in Demo 1.

Adventure 2.

[Choosing two or more conditions to analyze.]

Choose a slightly more complicated analysis to undertake, for example, focusing on surgical interventions (HIP-KNEE & CABG) or heart-related conditions (HF, AMI, & CABG).
See the same recommendations and hints as Adventure 1.

Adventure 3.

[Continue analyzing the pneuomnia data.]

Make sure to justify your choice.
Choose at least two new algorithms to perform. They can be unsupervised or supervised, but must be 2+.
A natural extension would be to do a Ridge and/or a LASSO; the former would tell you if you should be controlling for multicollinearity more than you currently are, the latter would tell you if you need to further shrink coefficients than you currently are. Additionally, you could consider a regression-based method that incorporates non-linearity, such as a random forest. However, you are free to choose anything you think is appropriate here!
Other choices I think you could potentially justify to stakeholders, if you were interested:

Switch to classification-based random forest using ComparedToNational_Hospital return days for pneumonia patients
kNN using the 2 clusters we’ve identified and named in our segmentation analysis
Our data are also ready for a neural network analysis (e.g., a feed-forward NN)!

Deliverables

I am looking for 1-2 markdown documents with their knitted HTML. For example, if you’re choosing Adventures 1 or 2, you may want to work through this document once, make a copy of this markdown document, move the source code to the TOP, edit the source code as needed, and then go through this again from the beginning using the new condition(s) you chose.

Alternatively, if you’re choosing Adventure 3, maybe you just choose to add on to the bottom of this document, replacing the ‘Choose Your Own Adventure’ section with your actual adventure.

Just make sure to answer the questions well,and make sure to justify the decisions you make. Tell me WHY you’re choosing the condition(s) you are in Adventures 1 or 2, or Tell me WHY you’re doing the analyses you are in Adventure 3. Other than that, make this your own learning adventure!

References

Box, G. E. P., and D. R. Cox. 2018. “An Analysis of Transformations.” Journal of the Royal Statistical Society: Series B (Methodological) 26 (2): 211–43. https://doi.org/10.1111/j.2517-6161.1964.tb00553.x.

De Alba, Israel, and Alpesh Amin. 2014. “Pneumonia Readmissions: Risk Factors and Implications.” Ochsner Journal 14 (4): 649–54.

R Core Team. 2019. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org.

Yeo, In‐Kwon, and Richard A. Johnson. 2000. “A New Family of Power Transformations to Improve Normality or Symmetry.” Biometrika 87 (4): 954–59. https://doi.org/10.1093/biomet/87.4.954.

Zou, Hui, and Trevor Hastie. 2005. “Regularization and Variable Selection via the Elastic Net.” Journal of the Royal Statistical Society Series B: Statistical Methodology 67 (2): 301–20. https://doi.org/10.1111/j.1467-9868.2005.00503.x.

Project 1: Biomedical & Clinical Informatics

Merrimack College DSE6630: Healthcare & Life Sciences Analytics

Katherine S. Geist, PhD

20 May 2025

1 Introduction

1.1 Our Analysis Plan

1.2 Your objective.

Adventure 1.

Adventure 2.

Adventure 3.

1.3 A Fork in the Road!

Option 1.3.1

Option 1.3.2

Question 1: [1 point]

1.4 Helper code: Using R like an object-oriented language

2 Exploratory Data Analysis: Which of the possible target (outcome) variables should we use? (Continued)

2.1 Overview of our options

2.2 Summary statistics of the possible targets

2.3 Comparing the four possible metrics toComparedToNational_Hospital return days for pneumonia patients

2.3.1 Example with geom_violin():

Question 2A: [0.5 points]

Question 2B: [0.5 points]

2.3.2 Example with geom_density():

Question 3: [0.5 points]

2.4 What is the relationship of pneumonia-related hospital readmissions to the hospital performance metrics?

Question 4: [1 point]

2.5 What is the relationship of pneumonia-related hospital readmissions to the patient satisfaction metrics?

Question 5: [1 point]

Question 6: [2 points]

2.5 Final remarks on target selection.

Question 7: [1 point]

2.6 Bias and Limitation

Question 8: [2 points]

3 Pre-processing and Feature Selection

3.1 Further feature selection and refinement.

3.1.1 Dropping the targets we aren’t planning to use. [Manual Method]

Question 9A: [0.5 points]

3.1.2 Dropping any features with near-zero variance. [Automated Method]

Question 9B: [0.5 points]

Question 9C: [0.5 points]

Question 9D: [1 point]

3.2 Assess missingness & devise an imputation plan

Question 10: [1 point]

3.2.1 Use the md.pairs() function from the mice package to assess the extent of missing values before imputing.

Question 11A: [0.5 points]

Question 11B: [0.5 points]

3.3 Drop rows that are missing from the target variable.

Question 12: [0.5 points]

3.4 Do the splits!

Question 13: [1 point]

Question 14A: [1.5 points]

Question 14B: [0.5 points]

3.5 Impute missing variables using missForest.

3.5.1 Running missForest

3.5.2 Load this if it will not run for you

3.5.3 Quick summary of the results

Question 15: [1 point]

Question 16: [1 point]

3.5.4 Putting it alllllll back together…

3.5.4.1 Put the original State and target variables back

3.5.4.2 Now do the imputation for test

3.5.4.3 Load this if it will not run for you

3.5.4.4 Put the original State and target variables back on test

3.5.4.5 Drop the features we determined we should not retain due to unstable imputation from BOTH train and test

Question 17: [1 point]

3.6 Another Fork in the Road

3.6.1 The Left Fork - No Encoding, Just Converting Data Back After missForest

3.6.2 The Right Fork - Frequency Encoding

Question 18: [1 point]

3.7 Transformation & Scaling

Question 19A: [1 point]

3.7.1 Box-Cox Transformation

3.7.1.1 There are multiple ways to apply the Box-Cox transform in R, but we are going to take advantage of the one in caret to make our lives easier, BoxCoxTrans().

3.7.1.2 Now, apply the Box-Cox transformation to the train data:

3.7.1.3 Assess how it did:

Figures 8a-b. Predicted Readmission Rate After Box-Cox Transformation

Question 19B: [1.5 points]

3.7.2 Centering & Scaling

Question 19C: [0.5 points]

3.7.3 To save time, you are welcome to start from here after you’ve worked through the above steps.

1.4 Helper code: Using `R` like an object-oriented language

2.3 Comparing the four possible metrics to`ComparedToNational_Hospital return days for pneumonia patients`

2.3.1 Example with `geom_violin()`:

2.3.2 Example with `geom_density()`:

3.2.1 Use the `md.pairs()` function from the `mice` package to assess the extent of missing values before imputing.

3.5 Impute missing variables using `missForest`.

3.5.1 Running `missForest`

3.5.4.1 Put the original `State` and `target` variables back

3.5.4.2 Now do the imputation for `test`

3.5.4.4 Put the original `State` and `target` variables back on `test`

3.5.4.5 Drop the features we determined we should not retain due to unstable imputation from BOTH `train` and `test`

3.6.1 The Left Fork - No Encoding, Just Converting Data Back After `missForest`

3.7.1.1 There are multiple ways to apply the Box-Cox transform in R, but we are going to take advantage of the one in `caret` to make our lives easier, `BoxCoxTrans()`.

3.7.1.2 Now, apply the Box-Cox transformation to the `train` data:

4.2.1 Fit a linear regression with the training data, using the Box-Cox transformed `Predicted Readmission Rate` as the target.

4.2.1 Next, let’s calculate the VIFs using the `vif()` function that is part of the `car` package.

4.3.2 Now drop all of the `ComparedToNational` and `PaymentCategory` features except `ComparedToNational_Hospital return days for pneumonia patients`

6.2.2.1 We will start by setting our CV parameters just as we did on the last project, using the `trainControl()` function in `caret`.